Scale-Invariant Minimum-Cost Curves: Fair and Robust Design Implements

Four functionals for the computation of minimum cost curves are compared. Minimization of these functionals result in the widely studied Minimum Energy Curve (MEC), the recently introduced Minimum Variation Curve (MVC), and their scale-invariant counterparts, (SI-MEC, SI-MVC). We compare the stability and fairness of these curves using a variety of simple interpolation problems. Previously, we have shown MVC to possess superior fairness. In this paper we show that while MVC have fairness and stability superior to MEC they are still not stable in all configurations. We introduce the SI-MVC as a stable alternative to the MVC. Like the MVC, circular and helical arcs are optimal shapes for the SI-MVC. Additionally, the application of scale invariance to functional design allows us to investigate locally optimal curves whose shapes are dictated solely by their topology, free of any external interpolation or arc length constraints.